If (1+i)2 2-i =x+iy, Then Find The Value Of X+y.​

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Introduction

Complex numbers are an extension of the real number system, which includes a new element, 'i', that satisfies the equation i^2 = -1. In this article, we will explore how to solve complex numbers and apply this knowledge to solve a specific problem. We will use the given equation (1+i)^2 = 2-i to find the value of x+y.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form a+bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. The real part of the complex number is 'a', and the imaginary part is 'b'. Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.

Solving the Given Equation

The given equation is (1+i)^2 = 2-i. To solve this equation, we need to expand the left-hand side using the binomial theorem.

Expanding the Left-Hand Side

Using the binomial theorem, we can expand (1+i)^2 as follows:

(1+i)^2 = 1^2 + 2(1)(i) + i^2 = 1 + 2i + i^2

Since i^2 = -1, we can substitute this value into the equation:

(1+i)^2 = 1 + 2i - 1 = 2i

Now, we can equate the left-hand side and the right-hand side of the equation:

2i = 2-i

Simplifying the Equation

To simplify the equation, we can multiply both sides by the conjugate of the right-hand side, which is 2+i.

2i(2+i) = (2-i)(2+i)

Expanding both sides, we get:

4i + 2i^2 = 4 + 2i - 2i - i^2

Simplifying further, we get:

4i - 2 = 4 - 1

Combine like terms:

4i - 2 = 3

Add 2 to both sides:

4i = 5

Finding the Value of x+y

Now that we have the value of 4i, we can find the value of x+y. Since 4i = 5, we can write:

x+iy = 5

Comparing this with the given equation x+iy = (1+i)^2, we can see that x = 0 and y = 5.

Conclusion

In this article, we solved the given equation (1+i)^2 = 2-i and found the value of x+y. We used the binomial theorem to expand the left-hand side and then simplified the equation by multiplying both sides by the conjugate of the right-hand side. Finally, we found the value of x+y by comparing the given equation with the expanded equation.

Key Takeaways

  • Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.
  • The binomial theorem can be used to expand expressions of the form (a+b)^n.
  • The conjugate of a complex number can be used to simplify equations.

Practice Problems

  1. Solve the equation (2-i)^2 = x+iy and find the value of x+y.
  2. Solve the equation (3+2i)(1-i) = x+iy and find the value of x+y.

Answer Key

  1. x = 5, y = -3
  2. x = 1, y = 5
    Frequently Asked Questions (FAQs) on Complex Numbers =====================================================

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form a+bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit.

Q: What is the imaginary unit 'i'?

A: The imaginary unit 'i' is a number that satisfies the equation i^2 = -1. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, (a+bi) + (c+di) = (a+c) + (b+d)i.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, (a+bi) - (c+di) = (a-c) + (b-d)i.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that i^2 = -1. For example, (a+bi)(c+di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can multiply the numerator and denominator by the conjugate of the denominator. For example, (a+bi)/(c+di) = ((a+bi)(c-di))/((c+di)(c-di)) = ((ac+bd) + (bc-ad)i)/c^2 + d^2.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number a+bi is a-bi.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use the fact that i^2 = -1 and the distributive property. For example, (a+bi)^2 = a^2 + 2abi + b2i2 = (a^2 - b^2) + 2abi.

Q: What is the modulus of a complex number?

A: The modulus of a complex number a+bi is the distance from the origin to the point (a,b) in the complex plane. It is given by |a+bi| = sqrt(a^2 + b^2).

Q: What is the argument of a complex number?

A: The argument of a complex number a+bi is the angle between the positive real axis and the line segment from the origin to the point (a,b) in the complex plane. It is given by arg(a+bi) = arctan(b/a).

Q: How do I solve equations involving complex numbers?

A: To solve equations involving complex numbers, you can use the same techniques as for solving equations involving real numbers, such as factoring, quadratic formula, and substitution.

Q: What are some common applications of complex numbers?

A: Complex numbers have many applications in mathematics, physics, engineering, and other fields, such as:

  • Electrical engineering: complex numbers are used to represent AC circuits and analyze their behavior.
  • Signal processing: complex numbers are used to represent signals and analyze their frequency content.
  • Control theory: complex numbers are used to analyze and design control systems.
  • Quantum mechanics: complex numbers are used to represent wave functions and analyze the behavior of particles.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

  • Forgetting to use the imaginary unit 'i' when working with complex numbers.
  • Not using the distributive property and the fact that i^2 = -1 when multiplying and dividing complex numbers.
  • Not simplifying complex expressions by combining like terms and using the fact that i^2 = -1.
  • Not using the conjugate of a complex number when dividing complex numbers.

Q: How can I practice working with complex numbers?

A: You can practice working with complex numbers by:

  • Solving problems involving complex numbers, such as adding, subtracting, multiplying, and dividing complex numbers.
  • Simplifying complex expressions and finding the modulus and argument of complex numbers.
  • Using complex numbers to solve equations and analyze systems.
  • Working with complex numbers in different contexts, such as electrical engineering, signal processing, and control theory.